Silviase/jgov_v1.4_full_ann huggingface dataset:
Writing $ \mu_{t}(h) $ the Lagrange multiplier associated to the capital accumulation function one gets:

$$ \begin{aligned}p_{I,t}\lambda_{t}(h)&=\mu_{t}(h)\varepsilon_{I,t}\left[1-\mathcal{S}\left(\frac{I_{t}(h)}{I_{t-1}(h)}\right)-\frac{I_{t}(h)}{I_{t-1}(h)}\mathcal{S}^{\prime}\left(\frac{I_{t}(h)}{I_{t-1}(h)}\right)\right]\\&+\beta\mathbb{E}_{t}\left[\mu_{t+1}(h)\varepsilon_{I,t+1}\left(\frac{I_{t+1}(h)}{I_{t}(h)}\right)^{2}\mathcal{S}^{\prime}\left(\frac{I_{t+1}(h)}{I_{t}(h)}\right)\right]\end{aligned} $$

$$ \mu_{t}(h)\delta^{\prime}\left(z_{t}(h)\right)=\lambda_{t}(h)r_{t}^{k} $$

and at last:

$$ \mu_{t}(h)=\beta\mathbb{E}_{t}\Big[\mu_{t+1}(h)\Big(1-\delta(z_{t+1}(h))\Big)+\lambda_{t+1}(h)r_{t+1}^{k}z_{t+1}(h)\Big] $$

Given the symmetrical nature of the solution for the household’s problem, we get the following aggregated relationships:

$$ \left(C_{t}-\eta C_{t-1}\right)^{-\sigma_{c}}\exp\left\{\varepsilon_{L,t}\frac{\sigma_{c}-1}{1+\sigma_{l}}L_{t}^{1+\sigma_{l}}\right\}=\lambda_{t} $$

$$ \lambda_{t}=\beta\varepsilon_{B,t}R_{t}\mathbb{E}_{t}\left[\frac{\lambda_{t+1}}{\pi_{t+1}}\right] $$

$$ u_{t}\varepsilon_{L,t}\left(\sigma_{c}-1\right)L_{t}^{\sigma_{l}}=-\lambda_{t}\frac{W_{t}^{m}}{P_{t}} $$

$$ \frac{p_{I,t}}{\varepsilon_{I,t}}=Q_{t}\left[1-\mathcal{S}\left(\frac{I_{t}}{I_{t-1}}\right)-\frac{I_{t}}{I_{t-1}}\mathcal{S}^{\prime}\left(\frac{I_{t}}{I_{t-1}}\right)\right] $$

$$ +\beta\mathbb{E}_{t}\left[\frac{\lambda_{t+1}}{\lambda_{t}}Q_{t+1}\frac{\varepsilon_{I,t+1}}{\varepsilon_{I,t}}\left(\frac{I_{t+1}}{I_{t}}\right)^{2}\mathcal{S}^{\prime}\left(\frac{I_{t+1}}{I_{t}}\right)\right] $$

$$ Q_{t}\delta^{\prime}\left(z_{t}\right)=r_{t}^{k} $$

$$ Q_{t}=\beta\mathbb{E}_{t}\left[\frac{\lambda_{t+1}}{\lambda_{t}}\left(Q_{t+1}\Big(1-\delta(z_{t+1})\Big)+r_{t+1}^{k}z_{t+1}\right)\right] $$

Here, $ Q_{t} \equiv \mu_{t}/\lambda_{t} $ is Tobin's Q.
### 1.2 Production

#### 1.2.1 Final good producers

Producers of final good, $ Y_{t} $ , operate in a perfectly competitive environment, assembling a continuum of diversified intermediary goods written $ Y_{t}(\iota) $ with $ \iota \in [0,1] $ . They have access to a unique constant return aggregation technology as in Kimball (1996), implicitly defined by

$$ \int_{0}^{1}\mathcal{G}_{f}\left(\frac{Y_{t}(\iota)}{Y_{t}}\right)\mathrm{d}\iota=1 $$

where $ G_{f} $ is a strictly increasing concave function such that $ \mathcal{G}_{f}(1)=1 $ . We follow Dotsey and King (2005) or Levin et al. (2007) and adopt the following functional form for this aggregation function:

$$ \begin{aligned}\mathcal{G}_{f}(x)=&\frac{\theta_{f}(1+\psi_{f})}{(1+\psi_{f})(\theta_{f}(1+\psi_{f})-1)}\Big[\big(1+\psi_{f}\big)x-\psi_{f}\Big]^{\frac{(1+\psi_{f})\theta_{f}-1}{(1+\psi_{f})\theta_{f}}}\\&-\left[\frac{\theta_{f}(1+\psi_{f})}{(1+\psi_{f})(\theta_{f}(1+\psi_{f})-1)}-1\right]\end{aligned} $$

Parameter $ \psi_{f} $ characterize the curvature of the demand function.

The producer of final good chooses the quantity of intermediary goods $ \iota $ so as to maximize her real profit:

$$ Y_{t}-\int_{0}^{1}\frac{P_{t}(\iota)}{P_{t}}Y_{t}(\iota)\mathrm{d}\iota $$

under the technological constraints (1.16) and (1.17). As the aggregation function is homogeneous of degree one, it is equivalent to minimize cost per unit with respect to the relative demand of intermediary good $ \iota $ under the technological constraint.

The first order condition of optimality determines the demand for intermediary good $ \iota $ :

$$ \frac{Y_{t}(\iota)}{Y_{t}}=\frac{1}{1+\psi_{f}}\left[\left(\frac{P_{t}(\iota)/P_{t}}{\Theta_{t}}\right)^{-(1+\psi_{f})\theta_{f}}+\psi_{f}\right] $$

where $ \Theta_{t} $ is the Lagrange multiplier associated with the technological constraints (1.16) and (1.17) for the representative firm. Substituting (1.18)
in the technological constraints, one gets the following expression for the Lagrange multiplier:

$$ \Theta_{t}=\left(\int_{0}^{1}\left(\frac{P_{t}(\iota)}{P_{t}}\right)^{1-\theta_{f}(1+\psi_{f})}\mathrm{d}\iota\right)^{\frac{1}{1-\theta_{f}(1+\psi_{f})}} $$

The price elasticity of demand is given by:

$$ \epsilon\left(\widetilde{Y}_{t}(\iota)\right)=-\frac{\mathcal{G}^{\prime}\left(\widetilde{Y}_{t}(\iota)\right)}{\widetilde{Y}_{t}(\iota)\mathcal{G}^{\prime \prime}\left(\widetilde{Y}_{t}(\iota)\right)} $$

and, with particular aggregation function adopted in this study,

$$ \epsilon\left(\widetilde{Y}_{t}(\iota)\right)=\theta_{f}\left[1+\psi_{f}-\frac{\psi_{f}}{\widetilde{Y}_{t}(\iota)}\right] $$

When $ \psi_{f} $ is equal to zero, we get back to the more usual case of the CES aggregator of Dixit and Stiglitz (1977) with a price elasticity of demand equal to $ \theta_{f} $ . More generally, one remarks that demand is more sensitive to price when the level of demand is important if and only if parameter $ \psi_{f} $ is positive. We expect therefore to obtain a negative value for this parameter.

Finally, as the final good sector is perfectly competitive, profit for the representative firm must be zero and we derive the aggregate price index:

$$ P_{t}=\frac{\psi_{f}}{1+\psi_{f}}\int_{0}^{1}P_{t}(\iota)\mathrm{d}\iota+\frac{1}{1+\psi_{f}}\left(\int_{0}^{1}P_{t}(\iota)^{1-(1+\psi_{f})\theta_{f}}\mathrm{d}\iota\right)^{\frac{1}{1-(1+\psi_{f})\theta_{f}}} $$

#### 1.2.2 Intermediary goods producers

A continuum of firms $ \iota\in[0,1] $ in monopolistic competition produce intermediary goods for the producers of the final good. These firms have all access to the same Cobb–Douglas technology in to transform physical capital and labor in differentiated intermediary goods:

$$ Y_{t}(\iota)=\left(K_{t}^{d}(\iota)\right)^{\alpha}\left(A_{t}L_{t}^{d}(\iota)\right)^{1-\alpha} $$
where $ K_{t}^{d}(\iota) $ and $ L_{t}^{d}(\iota) $ are demands of intermediary good firm $ \iota $ for physical capital, and labor, respectively; $ A_{t} $ is technical progress, neutral in Harrod sense. The latter term is further decomposed in a trend component $ A_{T,t} $ and a cyclical on $ A_{C,t} $ . We have then,

$$ \Delta\log\mathcal{A}_{T,t}\sim\mathrm{AR}(1)\mathrm{stationary with mean}\log(1+g) $$

$$ \log\mathcal{A}_{C,t}\sim\mathrm{AR}(1)\mathrm{stationary with zero mean}. $$

Each intermediary firm $ \iota\in[0,1] $ buys freely its production factors on competitive markets taking their price as given. The firm $ \iota\in[0,1] $ decides upon the mix of physical capital $ (K_{t}^{d}(\iota)) $ and labor $ (L_{t}^{d}(\iota)) $ so as to minimize its cost, $ r_{t}^{k}K_{t}^{d}(\iota)+w_{t}L_{t}^{d}(\iota) $ , under the technological constraint (1.21). The firm optimal behavior on the factor markets is summarized by the following factor prices frontier:

$$ \frac{w_{t}L_{t}^{d}(\iota)}{r_{t}^{k}K_{t}^{d}(\iota)}=\frac{1-\alpha}{\alpha} $$

where $ w_{t} \equiv W_{t}/P_{t} $ is the real wage. The ratio of capital to labor is invariant across firms. Using the factor prices frontier, we rewrite the total cost of firm $ \iota $ as a function of the stock of capital:

$$ C T_{t}(\iota)=\frac{r_{t}^{k}K_{t}^{d}(\iota)}{\alpha} $$

On the other hand, as the returns of scale is constant, we know that the total cost can also be written as

$$ C T_{t}(\iota)=m c_{t}(\iota)Y_{t}(\iota) $$

where $ mc_{t}(\iota) $ is the real marginal cost. We derive then the following expression for the marginal cost of firm $ \iota $ :

$$ m c_{t}(\iota)=A_{t}^{\alpha-1}\left(\frac{r_{t}^{k}}{\alpha}\right)^{\alpha}\left(\frac{w_{t}}{1-\alpha}\right)^{1-\alpha}\equiv m c_{t} $$

Again, marginal cost doesn’t depend on the size of the firm and is constant across firms.

The nominal profit of a firm that offers price P at date t is given by:

$$ \begin{aligned}\Pi_{t}(\mathcal{P})=&\left(\varepsilon_{y,t}\frac{\mathcal{P}}{P_{t}}-m c_{t}\right)\frac{P_{t}Y_{t}}{1+\psi_{f}}\times\\&\left[\left(\frac{\mathcal{P}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\left(\int_{0}^{1}\left(\frac{P_{t}(\iota)}{P_{t}}\right)^{1-(1+\psi_{f})\theta_{f}}\mathrm{d}f\right)^{\frac{(1+\psi_{f})\theta_{f}}{1-(1+\psi_{f})\theta_{f}}}+\psi_{f}\right]\end{aligned} $$
where $ \log \varepsilon_{y,t} $ , a zero mean AR stationary process, is a shock to the sales of the firm. This shock will show up in the Phillips curve.

Firm $ \iota $ has market power but can’t decide of the its optimal price in each period. Following a Calvo scheme, at each date, the firm receives a signal telling it whether it can revise its price $ P_{t}(\iota) $ in an optimal manner or not. There is a probability $ \xi_{p} $ that the firm can’t revise its price in a given period. In such a case, the firm follows the following rule:

$$ P_{t}(\iota)=[\bar{\pi}_{t}]^{\gamma_{p}}\left[\frac{P_{t-1}}{P_{t-2}}\right]^{1-\gamma_{p}}P_{t-1}(\iota)=\Gamma_{t}P_{t-1}(\iota) $$

where $ \bar{\pi}_{t} $ is the inflation target of the monetary authorities. More generally, we write

$$ \Gamma_{t+j,t}=\left(\prod_{h=0}^{j-1}\bar{\pi}_{t+h}\right)^{\gamma_{p}}\left(\prod_{h=0}^{j-1}\pi_{t+h}\right)^{1-\gamma_{p}}=\Gamma_{t+1}\Gamma_{t+2}\cdots\Gamma_{t+j} $$

the growth factor of the price of a firm that doesn’t receive a favorable signal during j successive periods(for j = 0 we have $ \Gamma_{t,t} = 1 $ ; for j = 1, we have $ \Gamma_{t+1,t} = \Gamma_{t+1} $ ). When the firm $ \iota $ receives a positive signal (with probability $ 1 - \xi_{p} $ ), it chooses price $ P_{t}(\iota) $ that maximizes its profit.

Let $ \widetilde{\mathcal{V}}_{t} $ be the value of a firm that receives a positive signal in period t and $ \mathcal{V}_{t}(P_{t-1}(\iota)) $ the value of a firm that receives a negative signal. As a firm that receives a negative signal follows simply the ad hoc pricing rule (1.25), its value at time t depends only on $ P_{t-1}(\iota) $ . For a firm that receives a positive signal, its value at period t is

$$ \widetilde{\mathcal{V}}_{t}=\max_{\mathbf{P}}\Biggl\{\Pi_{t}\Big(\mathbf{P}\Big)+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{p})\widetilde{\mathcal{V}}_{t+1}+\xi_{p}\mathcal{V}_{t+1}\big(\mathbf{P}\big)\right)\right]\Biggr\} $$

where $ \Lambda_{t} $ is the Lagrange multiplier of the budget constraint of the representative household and $ P_{t}\Lambda_{t}=\lambda_{t} $ . Let $ P^{\star} $ be the optimal price chosen by the firm that can re-optimize. The value of a firm that can't re-optimize is

$$ \begin{aligned}\mathcal{V}_{t}\big(P_{t-1}(\iota)\big)=&\Pi_{t}\Big(\Gamma_{t}P_{t-1}(\iota)\Big)\\+&\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{p})\widetilde{\mathcal{V}}_{t+1}+\xi_{p}\mathcal{V}_{t+1}\big(\Gamma_{t}P_{t-1}(\iota)\big)\right)\right]\end{aligned} $$

The first order condition and the envelope theorem give

$$ \Pi_{t}^{\prime}\big(P^{\star}\big)+\beta\xi_{p}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{V}_{t+1}^{\prime}\big(P^{\star}\big)\right]=0 $$
$$ \frac{\mathcal{V}_{t}^{\prime}\big(P_{t-1}(\iota)\big)}{\Gamma_{t}}=\Pi_{t}^{\prime}\big(\Gamma_{t}P_{t-1}(\iota)\big)+\beta\xi_{p}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{V}_{t+1}^{\prime}\big(\Gamma_{t}P_{t-1}(\iota)\big)\right] $$

with the derivative of profit at P :

$$ \begin{align*}\Pi_{t}^{\prime}\big(\mathcal{P}\big)\quad=&\quad\varepsilon_{y,t}\frac{1-\theta_{f}(1+\psi_{f})}{1+\psi_{f}}\left(\frac{\mathcal{P}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\Theta_{t}^{(1+\psi_{f})\theta_{f}}Y_{t}\\&+\theta_{f}\left(\frac{\mathcal{P}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}-1}\Theta_{t}^{(1+\psi_{f})\theta_{f}}mc_{t}Y_{t}+\frac{\psi_{f}}{1+\psi_{f}}\varepsilon_{y,t}Y_{t}\end{align*} $$

Let’s write temporarily, in order to simplify notations, P, the price inherited from the past. One can rewrite, one period ahead

$$ \mathcal{V}_{t+1}^{\prime}\big(\mathcal{P}\big)=\Gamma_{t+1,t}\Pi_{t+1}^{\prime}\big(\Gamma_{t+1,t}\mathcal{P}\big)+\beta\xi_{p}\Gamma_{t+1,t}\mathbb{E}_{t+1}\left[\frac{\Lambda_{t+2}}{\Lambda_{t+1}}\mathcal{V}_{t+2}^{\prime}\big(\Gamma_{t+1,t}\mathcal{P}\big)\right] $$

Iterating toward the future and applying conditional expectation at time t, one gets

$$ \mathbb{E}_{t}\left[\mathcal{V}_{t+1}^{\prime}\big(\mathcal{P}\big)\right]=\mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Gamma_{t+1+j,t}\frac{\Lambda_{t+1+j}}{\Lambda_{t+1}}\Pi_{t+1+j}^{\prime}\big(\Gamma_{t+1+j,t}\mathcal{P}\big)\right] $$

By substitution $ \left(\mathcal{P}=P^{\star}\right) $ in the first order condition, one gets the following condition for the price chosen by the firm that gets a positive signal:

$$ \mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Gamma_{t+j,t}\frac{\Lambda_{t+j}}{\Lambda_{t}}\Pi_{t+j}^{\prime}\big(\Gamma_{t+j,t}P_{t}^{\star}\big)\right]=0 $$

One can get a more explicit expression for the price that satisfies equation (1.30). Substituting in this equation the expression of marginal profit (1.29) and dividing by $ P_{t}^{\star} - (1 + \psi_{f}) \theta_{f} $ one gets:

$$ \frac{P_{t}^{\star}}{P_{t}}=\frac{\theta_{f}(1+\psi_{f})}{\theta_{f}(1+\psi_{f})-1}\frac{\mathcal{Z}_{1,t}}{\mathcal{Z}_{2,t}}+\frac{\psi_{f}}{\theta_{f}(1+\psi_{f})-1}\left(\frac{P_{t}^{\star}}{P_{t}}\right)^{1+(1+\psi_{f})\theta_{f}}\frac{\mathcal{Z}_{3,t}}{\mathcal{Z}_{2,t}} $$

with

$$ \mathcal{Z}_{1,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{p})^{j}\lambda_{t+j}\left(\frac{\Gamma_{t+j}}{P_{t+j}/P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\Theta_{t+j}^{(1+\psi_{f})\theta_{f}}m c_{t+j}Y_{t+j} $$

$$ \mathcal{Z}_{2,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{p})^{j}\lambda_{t+j}\varepsilon_{y,t+j}\left(\frac{\Gamma_{t+j}}{P_{t+j}/P_{t}}\right)^{1-(1+\psi_{f})\theta_{f}}\Theta_{t+j}^{(1+\psi_{f})\theta_{f}}Y_{t+j} $$
$$ \mathcal{Z}_{3,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{p})^{j}\lambda_{t+j}\varepsilon_{y,t+j}\frac{\Gamma_{t+j}}{P_{t+j}/P_{t}}Y_{t+j} $$

writing $ P_{t+j}/P_{t} $ , the inflation factor between t and $ t+j $ , can be written equivalently $ \Pi_{i=1}^{j}\pi_{t+i} $ , and we can represent variables $ X_{1,t} $ , $ X_{2,t} $ et $ X_{3,t} $ in recursive form:

$$ \mathcal{Z}_{1,t}=\widehat{\lambda}_{t}m c_{t}\Theta_{t}^{(1+\psi_{f})\theta_{f}}\widehat{Y}_{t}+\beta\xi_{p}\mathbb{E}_{t}\left[\left(\frac{\pi_{t+1}}{\pi_{t-1}^{\gamma_{p}}\bar{\pi}_{t}^{1-\gamma_{p}}}\right)^{(1+\psi_{f})\theta_{f}}\mathcal{Z}_{1,t+1}\right] $$

$$ \mathcal{Z}_{2,t}=\widehat{\lambda}_{t}\varepsilon_{y,t}\Theta_{t}^{(1+\psi_{f})\theta_{f}}\widehat{Y}_{t}+\beta\xi_{p}\mathbb{E}_{t}\left[\left(\frac{\pi_{t+1}}{\pi_{t-1}^{\gamma_{p}}\bar{\pi}_{t}^{1-\gamma_{p}}}\right)^{(1+\psi_{f})\theta_{f}-1}\mathcal{Z}_{2,t+1}\right] $$

$$ \mathcal{Z}_{3,t}=\widehat{\lambda}_{t}\varepsilon_{y,t}\widehat{Y}_{t}+\beta\xi_{p}\mathbb{E}_{t}\left[\left(\frac{\pi_{t-1}^{\gamma_{p}}\bar{\pi}_{t}^{1-\gamma_{p}}}{\pi_{t+1}}\right)\mathcal{Z}_{3,t+1}\right] $$

Writing $\vartheta_{f,t} \equiv \int_{0}^{1} \frac{P_{t}(\iota)}{P_{t}} \mathrm{d}\iota$ can be written in recursive form:

$$ \vartheta_{f,t}=(1-\xi_{p})\frac{P_{t}^{\star}}{P_{t}}+\xi_{p}\frac{\bar{\pi}_{t}^{1-\gamma_{p}}\pi_{t-1}^{\gamma_{p}}}{\pi_{t}}\vartheta_{f,t-1} $$

we can finally write the equation $ (1.20) $ equivalently as:

$$ \frac{\psi_{f}\vartheta_{f,t}}{1+\psi_{f}}+\frac{\Theta_{t}}{1+\psi_{f}}=1 $$

In the end, inflation dynamics are characterized by equations $ (1.35) $ , $ (1.34) $ , $ (1.31) $ , $ (1.33a) $ , $ (1.33b) $ , $ (1.33c) $ .

### 1.3 Labor

Homogeneous labor $ L_{t} = \int_{0}^{1} L_{t}(h) \, \mathrm{d}h $ provided by the households is differentiated by a continuum of unions, $ \varsigma \in [0,1] $ . We have then $ L_{t} = \int_{0}^{1} l_{t}(\varsigma) \, \mathrm{d}\varsigma $ . Unions sell differentiated labor, $ l_{t}(\varsigma) $ , to an employment agency that aggregate different types of labor to offer it as input to the firms of the intermediary good sector. Unions have monopolistic power and the employment agency operates in a perfectly competitive manner.
#### 1.3.1 Employment agency

It aggregates labor $ l_{t}(\varsigma) $ provided by unions with an aggregation function as in Kimball (1996), defined implicitly by

$$ \int_{0}^{1}\mathcal{G}_{s}\left(\frac{l_{t}(\varsigma)}{\mathcal{L}_{t}}\right)\mathrm{d}\varsigma=1 $$

where $ G_{s} $ , a strictly increasing concave function such that $ \mathcal{G}_{s}(1)=1 $ , is defined as $ G_{f} $ in section 1.2 (replacing $ \theta_{f} $ by $ \theta_{s} $ and $ \psi_{f} $ by $ \theta_{s} $ ). The employment agency chooses the relative quantity of labor of type $ \varsigma $ such as minimizing the cost of production by unit of homogeneous labor, $ \frac{W_{t}(\varsigma)}{W_{t}}\frac{l_{t}(\varsigma)}{L_{t}} $ , under the technological constraint (1.36). The first order condition associated to the optimization program of the employment agency determines its demand of differentiated labor $ ^{1} $ $ \varsigma $ :

$$ \frac{l_{t}(\varsigma)}{\mathcal{L}_{t}}=\frac{1}{1+\psi_{s}}\left[\left(\frac{W_{t}(\varsigma)/W_{t}}{\Upsilon_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\psi_{s}\right] $$

where $ \Upsilon_{t} $ is the Lagrange multiplier associated with the technological constraint (1.36). Substituting (1.37) in the technological constraint, one gets the following expression for the Lagrange multiplier

$$ \Upsilon_{t}=\left(\int_{0}^{1}\left(\frac{W_{t}(\varsigma)}{W_{t}}\right)^{1-\theta_{s}\left(1+\psi_{s}\right)}\mathrm{d}\varsigma\right)^{\frac{1}{1-\theta_{s}\left(1+\psi_{s}\right)}} $$

As the employment agency behaves in a competitive manner, its profit is zero and we get the aggregate wage as

$$ W_{t}=\frac{\psi_{s}}{1+\psi_{s}}\int_{0}^{1}W_{t}(\varsigma)\mathrm{d}\varsigma+\frac{1}{1+\psi_{s}}\left(\int_{0}^{1}W_{t}(\varsigma)^{1-(1+\psi_{s})\theta_{s}}\mathrm{d}\varsigma\right)^{\frac{1}{1-(1+\psi_{s})\theta_{s}}} $$

#### 1.3.2 Unions

Unions supply differentiated labor services from the homogeneous labor supply from the households. Unions have market power because of this differentiation of labor services. We write the profit of a union offering wage $ W_{t}(\varsigma) $ and $ l_{t}(\varsigma) $ units of labor:

$$ \mathcal{S}_{t}\left(W_{t}(\varsigma)\right))=\left(\varepsilon_{l,t}W_{t}(\varsigma)-W_{t}^{m}\right)l_{t}(\varsigma) $$
where $ \log \varepsilon_{l,t} $ , is an exogenous shock on union's gains. It is an AR(1) process with zero mean. For a given demand, when $ \varepsilon_{l,t}=1 $ , the profit of the union is given by the difference between the wage asked to the employment agency and the wage paid to the household. By substitution in the demand function of the employment agency (1.37):

$$ \mathcal{S}_{t}\left(W_{t}(\varsigma)\right)=\left(\varepsilon_{l,t}W_{t}(\varsigma)-W_{t}^{m}\right)\frac{\mathcal{L}_{t}}{1+\psi_{s}}\left[\left(\frac{W_{t}(\varsigma)/W_{t}}{\Upsilon_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\psi_{s}\right] $$

Each union is subject to a Calvo lottery. In each period, a union can adjust the wage $ W_{t}(\varsigma) $ in an optimal manner with probability $ \xi_{w} $ . In this case, the union chooses wage $ W_{t}^{\star} $ that maximizes its profit knowing that in the future it may not have the opportunity to readjust the wage for some periods. When the lottery draw is negative for the union (with probability $ 1 - \xi_{w} $ ), it adjusts wages according to the following ad hoc rule:

$$ W_{t}(\varsigma)=\frac{\mathcal{A}_{T,t}}{\mathcal{A}_{T,t-1}}\bar{\pi}_{t-1}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}}W_{t-1}(\varsigma) $$

We write $ \Omega_{t} = (\mathcal{A}_{T,t}/\mathcal{A}_{T,t-1})\bar{\pi}_{t-1}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}} \equiv \Omega_{t,t-1} $ the growth factor of nominal wage asked by the union $ \varsigma $ at date t when this one doesn’t have the opportunity of revising it in an optimal manner. In this case, the union changes the wage by indexing it on (i) a convex mix of the inflation target of the monetary authority and of past inflation and (ii) the efficiency growth in the intermediary goods sector. We write

$$ \Omega_{t+j,t}=\frac{\mathcal{A}_{T,t+j}}{\mathcal{A}_{T,t}}\left(\prod_{h=0}^{j-1}\bar{\pi}_{t+h}\right)^{\gamma_{w}}\left(\prod_{h=0}^{j-1}\pi_{t+h}\right)^{1-\gamma_{w}}=\Omega_{t+1}\Omega_{t+2}\cdots\Omega_{t+j} $$

the growth factor of the wage of a union that gets negative signals during the next j periods (for j = 0, we have $ \Omega_{t,t} = 1 $ , for j = 1, we have $ \Omega_{t+1,t} = \Omega_{t+1} $ ).

Let $ \mathcal{U}_{t} $ be the value of a union that receives a positive signal at date t and $ \mathcal{U}_{t}(W_{t-1}(\varsigma)) $ the value of a union that receives the negative signal. In the latter case, the union follows simply the ad hoc rule (1.41), this explains why its value at date t depends upon $ W_{t-1}(\varsigma) $ . On the opposite, the optimization program of a union that receives a positive signal is purely turn towards the future. As unions have the same expectations about the future, they all choose the same optimal wage $ (W_{t}^{\star}) $ . More formally, the value at date t of
a union that receives a positive signal is

$$ \widetilde{\mathcal{U}}_{t}=\underset{\mathbf{W}}{\max}\Big\{\mathcal{S}_{t}\Big(\mathbf{W}\Big)+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{w})\widetilde{\mathcal{U}}_{t+1}+\xi_{w}\mathcal{U}_{t+1}\big(\mathbf{W}\big)\right)\right]\Big\} $$

where $ \Lambda_{t} $ is the Lagrange multiplier associated to the nominal budget constraint of the representative household.

The value of a union that receives a negative signal is

$$ \begin{aligned}\mathcal{U}_{t}\big(W_{t-1}(\varsigma)\big)=&\mathcal{S}_{t}\Big(\Gamma_{t}W_{t-1}(\varsigma)\Big)\\&+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{w})\widetilde{\mathcal{U}}_{t+1}+\xi_{w}\mathcal{U}_{t+1}\big(\Gamma_{t}W_{t-1}(\varsigma)\big)\right)\right]\end{aligned} $$

The first order condition and the application of the envelop theorem give

$$ \mathcal{S}_{t}^{\prime}\big(W_{t}^{\star}\big)+\beta\xi_{w}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{U}_{t+1}^{\prime}\big(W_{t}^{\star}\big)\right]=0 $$

$$ \frac{\mathcal{U}_{t}^{\prime}\big(W_{t-1}(\varsigma)\big)}{\Gamma_{t}}=\mathcal{S}_{t}^{\prime}\big(\Gamma_{t}W_{t-1}(\varsigma)\big)+\beta\xi_{p}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{U}_{t+1}^{\prime}\big(\Gamma_{t}W_{t-1}(\varsigma)\big)\right] $$

with the derivative of the union profit at W :

$$ \begin{align*}\mathcal{S}_{t}^{\prime}\big(\mathcal{W}\big)\quad=&\quad\varepsilon_{l,t}\frac{1-\theta_{s}(1+\psi_{s})}{1+\psi_{s}}\left(\frac{\mathcal{W}}{W_{t}}\right)^{-(1+\psi_{s})\theta_{s}}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}\mathcal{L}_{t}\\&+\theta_{s}\left(\frac{\mathcal{W}}{W_{t}}\right)^{-(1+\psi_{s})\theta_{s}-1}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}\frac{W_{t}^{m}}{W_{t}}\mathcal{L}_{t}+\frac{\psi_{s}}{1+\psi_{s}}\varepsilon_{l,t}\mathcal{L}_{t}\end{align*} $$

Let’s write temporarily, in order to simplify notations, W, the price inherited from the past. One can rewrite, one period ahead

$$ \mathcal{U}_{t+1}^{\prime}\big(\mathcal{W}\big)=\Omega_{t+1,t}\mathcal{S}_{t+1}^{\prime}\big(\Omega_{t+1,t}\mathcal{W}\big)+\beta\xi_{p}\Omega_{t+1,t}\mathbb{E}_{t+1}\left[\frac{\Lambda_{t+2}}{\Lambda_{t+1}}\mathcal{U}_{t+2}^{\prime}\big(\Omega_{t+1,t}\mathcal{W}\big)\right] $$

iterating toward the future and applying conditional expectation gives

$$ \mathbb{E}_{t}\left[\mathcal{U}_{t+1}^{\prime}\big(\mathcal{W}\big)\right]=\mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Omega_{t+1+j,t}\frac{\Lambda_{t+1+j}}{\Lambda_{t+1}}\mathcal{S}_{t+1+j}^{\prime}\big(\Omega_{t+1+j,t}\mathcal{W}\big)\right] $$

Substituting in the first order condition (pour $ W = W_{t}^{\star} $ ), one gets the following condition for an optimal wage choice by a union that receives a positive signal:

$$ \mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Omega_{t+j,t}\frac{\Lambda_{t+j}}{\Lambda_{t}}\mathcal{S}_{t+j}^{\prime}\big(\Omega_{t+j,t}W_{t}^{\star}\big)\right]=0 $$